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HF_cc-pvqz/pes_CIo3.dat
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Manuscript/seniority.tex
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\newcommand{\LCPQ}{Laboratoire de Chimie et Physique Quantiques (UMR 5626), Universit\'e de Toulouse, CNRS, UPS, France} \newcommand{\LCPQ}{Laboratoire de Chimie et Physique Quantiques (UMR 5626), Universit\'e de Toulouse, CNRS, UPS, France}
\title{Configuration interaction with seniority number and excitation degree} \title{Configuration Interaction with Seniority Number and Excitation Degree}
\author{F\'abris Kossoski} \author{F\'abris Kossoski}
\email{fkossoski@irsamc.ups-tlse.fr} \email{fkossoski@irsamc.ups-tlse.fr}
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% Abstract % Abstract
\begin{abstract} \begin{abstract}
%aimed at recovering both static and dynamic correlation, %aimed at recovering both static and dynamic correlation,
Here we propose a novel partitioning of the Hilbert space, hierarchy configuration interaction (hCI), We propose a novel partitioning of the Hilbert space, hierarchy configuration interaction (hCI),
where the degree of excitation (with respect to a given reference) and the seniority number (number of unpaired electrons) are combined in a single hierarchy parameter. where the degree of excitation (with respect to a given reference) and the seniority number (number of unpaired electrons) are combined in a single hierarchy parameter.
The key appealing feature of hCI is that it includes all classes of determinants that share the same scaling with the number of electrons and basis functions. The key appealing feature of hCI is that each level of the hierarchy accounts for all classes of determinants that share the same scaling with the system size.
In this way, it accounts for low-seniority high-excitation determinants lacking in excitation-based CI, while keeping the same computational scaling with system size. %number of electrons and basis functions.
By surveying the dissociation of multiple molecular systems, we examined how fast hCI and their excitation-based and seniority-based parents converge as %In this way, it accounts for low-seniority high-excitation determinants lacking in excitation-based CI, while keeping the same computational scaling with system size.
we step up towards the exact full CI limit. By surveying the dissociation of multiple molecular systems, we found that the overall performance of hCI usually exceeds or at least parallels that of excitation-based CI.
We found that the overall performance of hCI usually exceeds or at least parallels that of excitation-based CI. %By surveying the dissociation of multiple molecular systems, we examined how fast hCI and their excitation-based and seniority-based parents converge as we step up towards the exact full CI limit.
For small systems and basis sets, doubly-occupied CI (the first level of seniority-based CI) often remains the best option. %The overall performance of hCI usually exceeds or at least parallels that of excitation-based CI.
However, for larger systems or basis sets, and for higher accuracy, seniority-based CI becomes impractical. %For small systems and basis sets, doubly-occupied CI (the first level of seniority-based CI) often remains the best option, but becomes impractical for larger systems or basis sets, and for higher accuracy.
However, some of its interesting features, particularly the small non-parallelity errors, are partially recovered with hCI, at only a polynomical cost. %However, for larger systems or basis sets, and for higher accuracy, seniority-based CI becomes impractical.
We have futher explored the role of optimizing the orbitals at several levels of CI. %However, some of its interesting features, particularly the small non-parallelity errors, are partially recovered with hCI, at only a polynomial cost.
%We have further explored the role of optimizing the orbitals at several levels of CI.
For higher orders of hCI and excitation-based CI, For higher orders of hCI and excitation-based CI,
the additional computational burden and other known issues related to orbital optimization usually do not compensate the marginal improvements often observed, the additional computational burden related to orbital optimization usually do not compensate the marginal improvements compared with results obtained with Hartree-Fock orbitals.
when compared with results obtained with canonical Hartree-Fock orbitals.
The exception is orbital-optimized CI with single excitations, a minimally correlated model displaying the qualitatively correct description of single bond breaking, The exception is orbital-optimized CI with single excitations, a minimally correlated model displaying the qualitatively correct description of single bond breaking,
at a very modest computational cost. at a very modest computational cost.
%\bigskip %\bigskip
@ -191,7 +191,7 @@ at the same time as static correlation, by moving down (increasing the seniority
The second justification is computational. The second justification is computational.
%Fig.~\ref{fig:scaling} also illustrates how the number of determinants within each block scales with the number of occupied orbitals $O$ and the number of virtual orbitals $V$. %Fig.~\ref{fig:scaling} also illustrates how the number of determinants within each block scales with the number of occupied orbitals $O$ and the number of virtual orbitals $V$.
In the hCI class of methods, each next level of theory accomodates additional determinants from different excitation-seniority sectors (each block of Fig.~\ref{fig:allCI}). In the hCI class of methods, each level of theory accomodates additional determinants from different excitation-seniority sectors (each block of Fig.~\ref{fig:allCI}).
The key realization behind hCI is that the number of additional determinants presents the same scaling with respect to $N$, for all excitation-seniority sectors entering at a given hierarchy $h$. The key realization behind hCI is that the number of additional determinants presents the same scaling with respect to $N$, for all excitation-seniority sectors entering at a given hierarchy $h$.
%to $O$ and $V$, for all excitation-seniority sectors of a given hierarchy $h$. %to $O$ and $V$, for all excitation-seniority sectors of a given hierarchy $h$.
%This computational realization represents the second justification for the introduction of the hCI method. %This computational realization represents the second justification for the introduction of the hCI method.